## Reasoning from Data and Chance Summary

### Monday - Friday, July 2 - 6, 2012

The goal for every teacher in this working group is to create some resource that they (and other teachers) can use with students or teachers. In addition, we want participants to deepen their understanding of the fundamental components of a good statistical study:

1. The creation of an appropriate question.
2. Designing a plan for collecting data.
3. Analyzing and interpreting the results of data collection.
4. Making conclusions beyond the data.

We strated with a simple prompt: "Let's think about some Field Day events from school...

Which ones could we replicate here at PCMI?
What research questions might we ask?"

The group quickly converged on a research question involving cup-stacking:
"Does practice improve a math teachers' ability to stack cups?"

Within the first two days of class, the seven participants went deep into a lot of really important fundamental issues in statistics / data: They designed a protocol to run the study, collected data, and constructed a variety of simulations to determine if the results could have plausibly occurred by chance variation. Here are tome things they discussed:

• The importance of creating a clear, well defined research question
• The need for specific, repeatable clear protocols to answer your research question
• Designing protocols and controls on the measurement variables to reduce bias and variation in the results
• Choosing appropriate, measures of our key variables: "amount of practice" and "cup stacking success"
• Choosing appropriate visual displays to see patterns in the data
• Describing data distributions correctly
• Defining appropriate variables to measure improvements (First stack time - Last stack time
• Deciding if the slope of a linear model predicting last time from first time
• Distinguishing between what happened in a sample and whether we can generalize to a larger population
• Using simulations as a tool to see if sample results are too large to be due to random chance
• Interpreting the results of a simulation
• Executing all of these tasks on a variety of different technology platform (Excel, Fathom, "by hand," R, Maple)

We didn't simply pose those two questions and let them go: We listened carefully to their conversations, but allowed the participants to work through and answer their own questions. They were extremely thoughtful, engaged, and successful at arriving at good answers and resolutions to their questions. Once in a while, we would be there to suggest a good resource, or push them in a more productive direction, but the conversations were deep, interesting and productive.

During the second half of the week, participants got experience crating simulations and exploring data with Fathom, a Software package designed specifically for the learning and teaching of statistics. In addition, participants were given some exercises and lessons would familiarize them with tasks that match the Common Core State Standards in Statistics and Probability.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.